Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,f^b=fg,e^c=eg46 conjugacy classes — view the full character table.
| maximal subgroup of | #132131, #132132, #132133, … |
| maximal normal subgroup of | #132131, #132132, #132133, … |
| maximal quotient of | #99156, #99157, #99158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1725);